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Full text of "Treatise On Analysis Vol-Ii"

4   BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS        335

(15.4.7)    Let E be a complex Hilbert space. Then the Banach algebra
A = ?() (15.1.6) is a star algebra with respect to the involution wh-w*
(11.5).

Hilbert-Schmidt Operators

(15.4.8)    Let E be an infinite-dimensional separable complex Hilbert space.
An operator u e ? (E) is said to be a Hilbert-Schmidt operator if, for one
choice of a Hilbert basis (#), the series with general term |lw(<2n)||2 is con-
vergent. If (&) is another Hilbert basis of E, then ||w(an)||2 =  \(u(an) I *m)l2

m

by Parseval's identity (6.5.2), and therefore the series  l|w(an)||2 converges

n

if and only if the  double family   (\(u(an)\bm)\2)  *s   summable   (5.3.4).

by Parseval's identity. Hence an equivalent condition is that the series with
general term || u*(bn)\\2 should be convergent. It follows that, if w is a Hilbert-
Schmidt operator, the series with general term ||w(^)||2 converges for all
choices of the Hilbert basis (#), that w* is a Hilbert-Schmidt operator, and
that the number

(15.4.8.1)                       Nl2 = (III(a)lrl      =II"*II2

is independent of the Hilbert basis (an) chosen. We shall show that the set
J*?2(E) of Hilbert-Schmidt operators is an algebra, on which \\u\\2 is a norm,
and that, endowed with this norm and the involution u\u*, the algebra
j*?2(E) is a Banach algebra with involution, without unit element.

The fact that w is a Hilbert-Schmidt operator also means that, if (an) is
any Hilbert basis of E, the sequence (||w(#n)||) eRN belongs to the Hilbert
space /R (6.5), and the norm of this sequence in / is equal to ||w||2. If
u, i?eJ$?2(E), then the inequality \\(u + v)(an)\\ <jj \\u(an)\\ + \\v(an)\\ and the
preceding remark show that u H- v e J^2(E) and that \\u + v\\2 ^ \\u\\2 + ||u||2.
Clearly also Aw e &2(E) for all A e C, and ||Aw||2 = |A|  \\tt\\2.

For every continuous operator w e J(E), the operator w o u is a Hilbert-
Schmidt operator, by reason of the inequality

l|H<w(0B))|| < ||w||  \\u(an)\\
(5.5.1), and we have

(15.4.8.2)                               l|woW||2g ||w||  ||w||2.1,